1. Section 9.1: Three Dimensional Coordinate Systems
Practice HW from Stewart Textbook (not to hand in)
p. 641 # 1, 2, 3, 7, 10, 11, 13, 14, 15b, 16

2. 3-D Coordinate Axes z y x

3. Points are located using ordered triples (x, y, z).

4. Example 1: Plot the points (1, 1, 1), (2, 5, 0),
(-2, 3, 4), (1, 1, -4), and (2, -5, 3).
Solution:

5. z y x

6. Note: The 3-D coordinate axes divides the
coordinate system into 3 distinct planes – the x-y
plane z = 0, the y-z plane x = 0, and the x-z plane
y = 0.

10. Suppose we are now given the two points and in 3D space.z y x

11. Then

12. Example 2: Find the distance and midpoint
between the points (2, 1, 4) and (6, 5, 2).
Solution:

14. Recall: An isosceles triangle is a triangle where
the lengths of two of its sides are equal. A right
triangle is a triangle with a 90 degree angle where
the sum of squares of the shorter sides equals
the square of the hypotenuse.

15. c a x x b
Icosceles Triangle
Right Triangle

16. Example 3: Find the lengths of the sides of the
triangle PQR if P = (1, -3, -2), Q = (5, -1, 2) and
R = (-1, 1, 2). Deterrmine if the resulting triangle
is an isosceles or a right triangle.
Solution: (In typewritten notes)

17. Standard Equation of a Spherez y x

18. Standard Equation of a Sphere
The standard equation of a sphere with radius r
and center is given by
In particular, if the center of the sphere is at the
origin, that is, if , then the
equation becomes

19. Fact: When the standard equation of a sphere is
expanded and simplify, we obtain the general
equation of a sphere

20. General Equation of a Sphere

21. Example 4: Find the standard and general
equation of a sphere that passes through the
point (2, 1, 4) and has center (4, 3, 3).
Solution:

24. Example 5: Find the center and radius of the
sphere
Solution: